Limiting chance a random map has five fixed points

Pick a function $f:\{1,\dots,n\}\to\{1,\dots,n\}$ uniformly at random from all $n^n$ such functions, and let $F_n$ count the points with $f(i)=i$. The limit $\lim_{n\to\infty}\mathbb{P}(F_n=5)$ can be written as $c/e$ for a constant $c$. What is $c$?

$c=\tfrac1{120}$. For a uniformly random function the values $f(i)$ are independent, so each $i$ is fixed with probability $\tfrac1n$ independently and $F_n\sim\text{Bin}(n,\tfrac1n)$. This converges to $\text{Poisson}(1)$, giving $\mathbb{P}(F_n=5)\to\tfrac{e^{-1}}{5!}=\tfrac{1}{120e}$.